| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.UniformCoverage |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| open scoped Topology |
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| theorem sqrt_div_positive {a L : ℝ} (hL : 0 < L) : |
| Real.sqrt a/L = Real.sqrt (a/L^2) := by |
| rw [Real.sqrt_div' a (sq_nonneg L),Real.sqrt_sq_eq_abs,abs_of_pos hL] |
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| theorem freshCoverageGain_div {n : ℕ} (hn : 0 < n) (b q L : ℝ) (hL : 0 < L) : |
| freshCoverageGain n b q/L = |
| (b^2*(n:ℝ)/L)*(1-1/(n:ℝ))/4-Real.log 2/L- |
| Real.sqrt (Real.pi^2*(b^2*(n:ℝ)/L)*((b^2*(n:ℝ)/L)*q^2/2+Real.log 8/L)/4)- |
| Real.sqrt (Real.pi^2*(b^2*(n:ℝ)/L)*(Real.log 2/L)/4) := by |
| have hn' : (n:ℝ) ≠ 0 := by exact_mod_cast hn.ne' |
| unfold freshCoverageGain |
| rw [sub_div,sub_div,sub_div,sqrt_div_positive hL,sqrt_div_positive hL] |
| have hterm : b^2*((n:ℝ)-1)/4/L = (b^2*(n:ℝ)/L)*(1-1/(n:ℝ))/4 := by |
| field_simp |
| rw [hterm] |
| change _-Real.sqrt ((2*(Real.pi^2*b^2*(n:ℝ)/8)*(b^2*(n:ℝ)*q^2/2+Real.log 8))/L^2)- |
| Real.sqrt ((2*(Real.pi^2*b^2*(n:ℝ)/8)*Real.log 2)/L^2) = _ |
| congr 2 <;> field_simp <;> ring_nf |
| |
| theorem freshCoverageGain_div_tendsto (L b : ℕ → ℝ) {v q : ℝ} |
| (hL : Tendsto L atTop atTop) |
| (hV : Tendsto (fun n => (b n)^2*(n:ℝ)/L n) atTop (𝓝 v)) : |
| Tendsto (fun n => freshCoverageGain n (b n) q/L n) atTop |
| (𝓝 (v/4-Real.sqrt (Real.pi^2*v^2*q^2/8))) := by |
| have hLi : Tendsto (fun n => (L n)⁻¹) atTop (𝓝 (0:ℝ)) := |
| tendsto_inv_atTop_zero.comp hL |
| have hN : Tendsto (fun n : ℕ => (n:ℝ)⁻¹) atTop (𝓝 (0:ℝ)) := |
| tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop (R := ℝ)) |
| have h2 : Tendsto (fun n => Real.log 2/L n) atTop (𝓝 (0:ℝ)) := by |
| simpa only [mul_zero,div_eq_mul_inv] using hLi.const_mul (Real.log 2) |
| have h8 : Tendsto (fun n => Real.log 8/L n) atTop (𝓝 (0:ℝ)) := by |
| simpa only [mul_zero,div_eq_mul_inv] using hLi.const_mul (Real.log 8) |
| have H := (((hV.mul ((tendsto_const_nhds (x := (1:ℝ))).sub hN)).div_const 4).sub h2).sub |
| ((hV.const_mul (Real.pi^2) |>.mul ((hV.mul_const (q^2) |>.div_const 2).add h8)).div_const 4).sqrt |
| have H' := H.sub ((hV.const_mul (Real.pi^2) |>.mul h2).div_const 4).sqrt |
| have HH : Tendsto (fun n => |
| ((b n)^2*(n:ℝ)/L n)*(1-1/(n:ℝ))/4-Real.log 2/L n- |
| Real.sqrt (Real.pi^2*((b n)^2*(n:ℝ)/L n)*(((b n)^2*(n:ℝ)/L n)*q^2/2+Real.log 8/L n)/4)- |
| Real.sqrt (Real.pi^2*((b n)^2*(n:ℝ)/L n)*(Real.log 2/L n)/4)) atTop |
| (𝓝 (v/4-Real.sqrt (Real.pi^2*v^2*q^2/8))) := by |
| convert H' using 1 <;> simp only [one_div,sub_zero,mul_one,mul_zero,zero_div,Real.sqrt_zero,add_zero] |
| congr 2 |
| ring_nf |
| apply HH.congr' |
| filter_upwards [hL.eventually (eventually_gt_atTop (0:ℝ)),eventually_gt_atTop (0:ℕ)] with n hLn hn |
| exact (freshCoverageGain_div hn (b n) q (L n) hLn).symm |
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| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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